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In (Cartan-Eilenberg 56, XV.7) axioms for certain systems of bigraded modules (“Cartan-Eilenberg systems”) are given which imply the existence of a spectral sequence converging to . See also (Switzer 75, section 15, 1-7).
A Cartan-Eilenberg system consists of modules for each , morphisms for all , , and boundary morphisms for all , such that
,
,
and commute,
there are long exact sequences .
Subject to convergence conditions…
The spectral sequence induced from , is defined by
In particular
For , , one has
The key example of such a system are the relative cohomology groups of a filtered topological space (see at spectral sequence of a filtered complex) for any generalized cohomology theory (Cartan-Eilenberg 56, XV.7, Example 2). That forms a suitable system of modules is the statement of the exact sequence for triples of any generalized cohomology theory.
This case of the Cartan-Eilenberg spectral sequence came to be known as the Atiyah-Hirzebruch spectral sequence.
For a Serre fibration over a CW-complex . And for a multiplicative reduced generalized (Eilenberg-Steenrod) cohomology theory.
define a Cartan-Eilenberg system by
(where ) for with the obvious maps for , .
The Cartan-Eilenberg spectral sequence of this Cartan-Eilenberg system is the Serre-Atiyah-Hirzebruch spectral sequence.
The corresponding long exact sequences take the form
Henri Cartan, Samuel Eilenberg, Homological algebra, Princeton Univ. Press (1956)
Robert Switzer, Algebraic Topology - Homotopy and Homology, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.
Last revised on February 29, 2016 at 16:58:55. See the history of this page for a list of all contributions to it.